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Algebraic covers: Field of moduli versus field of definition. (English) Zbl 0906.12001
Let \(B\) be a regular, projective, geometrically irreducible variety over a field \(K\), let \(X\to B\) be a finite cover of \(B\), generically unramified, with \(X\) normal. It is a priori defined over \(K^{\text{sep}}\), the separable closure of \(K\). Suppose that \(X\to B\) is isomorphic to each of its conjugates by \(\text{Gal} (K^{\text{sep}}/K)\). Then, it is natural to ask the following question: is \(X\to B\) defined over \(K\)? The answer is no in general. A counter-example is known (Couveignes-Granboulan), moreover, it was known that, for \(G\)-covers, i.e. for Galois covers with group \(G\), the obstruction is measured by one characteristic class in \(H^2(K,Z(G))\) (\(Z(G)\) is the center of \(G\)). The authors study the obstruction in the above context. They prove that the situation is more complicated: the obstruction is controlled by finitely many characteristic classes in \(H^2(K,Z(G))\), where in that case \(G\) is the group of the cover. The methods that they have used yield new concrete criteria to prove that a cover is defined over \(K\). A particularly beautiful application is the following theorem (conjectured by E. Dew): a \(G\)-cover is defined over \(\mathbb{Q}\) if and only if it is defined over \(\mathbb{Q}_p\) for all places \(p\) (including \(p=\infty\)).

MSC:
12F12 Inverse Galois theory
14H30 Coverings of curves, fundamental group
11G35 Varieties over global fields
11G25 Varieties over finite and local fields
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